The Galton--Watson process (GWP) is a discrete-time branching process model that provides a powerful tool for analyzing epidemic data and estimating key epidemiological parameters such as the basic reproduction number. When used with surveillance-based cluster size data, the GWP can also elicit information about the extent of transmission heterogeneity, even when each transmission process is not directly observable. When cluster size distribution data are available, the parameters that govern the transmission can be statistically inferred by using the probability mass function that corresponds to the observed cluster size data. For multi-type GWPs, however, real-world applications remain limited, possibly because of the absence of conceptually and practically straightforward approaches for deriving the closed-form solution of the final size distribution. In the present study, we propose a framework for computing the final size distribution of multi-type GWPs, using a method for the choice of the Cauchy integral contour. We provide examples of how our framework can be applied to both simulated data and real-world data of Middle East respiratory syndrome, and discuss potential pitfalls surrounding the identifiability of parameters for statistical inference when using likelihoods that are not conditioned on extinction.

翻译：Galton–Watson过程（GWP）是一种离散时间分支过程模型，为分析流行病数据和估计基本再生数等关键流行病学参数提供了有力工具。当与基于监测的聚类规模数据结合使用时，即便每个传播过程无法直接观测，GWP仍可揭示传播异质性的程度。当聚类规模分布数据可用时，可通过利用与观测聚类规模数据相对应的概率质量函数，对控制传播的参数进行统计推断。然而，对于多类型GWP，实际应用仍较为有限，这可能是由于缺乏概念上和实践上直接的方法来推导最终规模分布的闭合解。在本研究中，我们提出了一种计算多类型GWP最终规模分布的框架，采用了柯西积分围道的选择方法。我们提供了该框架在模拟数据和中东呼吸综合征真实数据中的应用示例，并讨论了在使用未以灭绝为条件的似然函数进行统计推断时，参数可识别性方面的潜在陷阱。